Chapter 1

The three main themes of this course are; Open Data, Open Science, and Data Science. Open Data allows for the sharing of data between researchers in all fields. The course will teach how to use open data sets from various sources and how to analyse them in different ways as well as how to intepret the data.

Open Science allows for researchers to show and have their data reproduced by others using the same openly available software tools and methods. This course will teach students how to use these openly available software tools and methods.

Data Science falls into the realm of statistics. This course involves coding, programming, modeling, visualizing and analysing.

I am very excited to learn R Progamming and be guided through these themes of Open Data, Open Science, and Data Science :)

https://github.com/Ericcoles/IODS-project

Regression and model validation

1. Reading and checking the data

Dataset with the variables gender, age, attitude,deep,stra,surfand points created by combining questions in thelearning2014` data, as defined in the datacamp exercises. Scaled all combination variables to the original scales by taking the mean. Excluded observations where the exam points variable is zero. The dataset has 166 observations and 7 variables.

learning2014<- read.csv("/Users/eric/Documents/GitHub/IODS-project/data/learning2014.csv")
str(learning2014)
## 'data.frame':    166 obs. of  7 variables:
##  $ gender  : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...
##  $ age     : int  53 55 49 53 49 38 50 37 37 42 ...
##  $ Attitude: int  37 31 25 35 37 38 35 29 38 21 ...
##  $ deep    : num  3.58 2.92 3.5 3.5 3.67 ...
##  $ stra    : num  3.38 2.75 3.62 3.12 3.62 ...
##  $ surf    : num  2.58 3.17 2.25 2.25 2.83 ...
##  $ points  : int  25 12 24 10 22 21 21 31 24 26 ...
dim(learning2014)
## [1] 166   7

Installed the packages ggplot2 and GGally in Rstudio

library(ggplot2)
library(GGally)

2.Graphical overview of the data and summaries of the variables in the data.

p1 <- ggpairs(learning2014, mapping = aes(col=gender,alpha=0.3), lower=list(combo = wrap("facethist", bins = 20)))
p1

Genders are shown in Blue(males) and Pink(females). One can see cleary that points and Attitude has the highest correlation values in both males and females.This suggests that one’s attidude could effect the points one receives. The lowest correlation values for women are seen in Attitude and age,0.00756, suggesting that attitude and age do not have a relationship among women in the study sample. However men had a much higher correlation value in Attiude and age,0.0415, suggesting that there is more of a relationship between Attitude and age among men than in women in the study sample.

3. Regression Model

Here I chose three variables as explanatory variables and fit a regression model where exam points is the target (dependent) variable. Showing a summary of the fitted model, commenting and interpreting the results. As well as explaining and interpreting the statistical test related to the model parameters.

regression_model <- lm(points ~ Attitude + stra + age, data = learning2014)
summary(regression_model)
## 
## Call:
## lm(formula = points ~ Attitude + stra + age, data = learning2014)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -18.1149  -3.2003   0.3303   3.4129  10.7599 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 10.89543    2.64834   4.114 6.17e-05 ***
## Attitude     0.34808    0.05622   6.191 4.72e-09 ***
## stra         1.00371    0.53434   1.878   0.0621 .  
## age         -0.08822    0.05302  -1.664   0.0981 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.26 on 162 degrees of freedom
## Multiple R-squared:  0.2182, Adjusted R-squared:  0.2037 
## F-statistic: 15.07 on 3 and 162 DF,  p-value: 1.07e-08

Attitude, stra, and age are the independent variables in the regression model and the variable points is the dependent variable. From the t value as well as the p-value Pr, it shows that the variables stra and age are not statistically significant. However, it is shown from the data analysis that the relationship between points and Attitude is significant from the P-value of 4.72e-09 and a high t-value of 6.191.

4.Fitted Regression Model

regression_model_fit <- lm(points ~ Attitude, data = learning2014)
summary(regression_model_fit)
## 
## Call:
## lm(formula = points ~ Attitude, data = learning2014)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -16.9763  -3.2119   0.4339   4.1534  10.6645 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 11.63715    1.83035   6.358 1.95e-09 ***
## Attitude     0.35255    0.05674   6.214 4.12e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared:  0.1906, Adjusted R-squared:  0.1856 
## F-statistic: 38.61 on 1 and 164 DF,  p-value: 4.119e-09

Both ‘stra’ and ‘age’ did not have a statistically significant relationship with the dependent variable, and in this model they were removed.

The Attitude variable had a high t-value of 6.214 and a significant p-value of 4.12e-09.

From the data it is shown that the Multiple R-squared is 0.1906 and the Adjusted R-squared value is 0.1856.

R-squared cannot be used to determine whether the coefficient estimates and predictions are biased, which is why you must assess the residual plots. Reference

R-squared does not indicate if a regression model provides an adequate fit to your data. A good model can have a low R2 value. On the other hand, a biased model can have a high R2 value.Reference

5.Graphical model validation

Residuals vs Fitted values, Normal QQ-plot and Residuals vs Leverage.

par(mfrow = c(2,2))
plot(regression_model_fit, which = c(1, 2, 5))

The Residuals vs. Fitted plot

It is a scatter plot of residuals on the y axis and fitted values (estimated responses) on the x axis. The plot is used to detect non-linearity, unequal error variances, and outliers.

The residuals “bounce randomly” around the 0 line. This suggests that the assumption that the relationship is linear is reasonable.

The residuals roughly form a “horizontal band” around the 0 line. This suggests that the variances of the error terms are equal.

Three of the residuals(145,56,35) “stands out” from the basic random pattern of residuals. This suggests that there are outliers.

The Normal Q-Q Plot

The quantile quantile (qq) plot is a graphical technique for determining if two data sets come from populations with a common distribution. A q-q plot is a plot of the quantiles of the first data set against the quantiles of the second data set.

These two batches do appear to have come from populations with a common distribution as the majority of the values fall on the same line. This also contains outliers (145,56,35)

The Residuals vs. Leverage Plot

An outlier is defined as an observation that has a large residual. This plot allows one to see if any outliers have a high residual value. The graph shows that all of the values have leverage less than 0.04, suggesting that this a good model.

“Chapter4”

Access the required packages

library(MASS)
library(GGally)
library(ggplot2)
library(tidyr)
library(corrplot)
## corrplot 0.84 loaded

Loaded the Boston dataset

data("Boston")

The Boston dataset shows housing values in suburbs of Boston with variables that are related to the value of housing such as per capita crime rate by town, the proportion of residential land zoned for lots over 25,000 sq.ft, the proportion of non-retail business acres per town, proportion of owner-occupied units built prior to 1940, full-value property-tax rate per $10,000, pupil-teacher ratio by town, median value of owner-occupied homes in $1000s.

Explored the Boston dataset

str(Boston)
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
dim(Boston)
## [1] 506  14
summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08204   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00

The Boston data frame has 506 rows and 14 columns.

Graphical Overview of the data.

calculate the correlation matrix and round it

cor_matrix<-cor(Boston)

Visualized the correlation matrix

The function corrplot function was used to can be used to visualize the correlations.

corrplot(cor_matrix, method="circle")

cor_matrix<-cor(Boston) %>% round(digits = 2)
corrplot(cor_matrix, method="circle", type="upper", cl.pos="b", tl.pos="d", tl.cex = 0.6)

Positive correlations are displayed in blue and negative correlations in red color. Color intensity and the size of the circle are proportional to the correlation coefficients. In the right side of the correlogram, the legend color shows the correlation coefficients and the corresponding colors. The corrplot shows the correlations between variables of the Boston dataset. From the plot see that that rad (index of accessibility to radial highways) and tax (full-value property-tax rate per $10000) have the strongest positive correlation between the variables. While dis (weighted mean of distances to five Boston employment centres) and nox (nitrogen oxides concentration), age (proportion of owner-occupied units built prior to 1940) and indux (proportion of non-retail business acres per town), medv (meidan value of owner-occupied homes in $1000) and lstat (lower status of the population) have very strong negative correlation. # Printed the correlation matrix

cor_matrix
##          crim    zn indus  chas   nox    rm   age   dis   rad   tax
## crim     1.00 -0.20  0.41 -0.06  0.42 -0.22  0.35 -0.38  0.63  0.58
## zn      -0.20  1.00 -0.53 -0.04 -0.52  0.31 -0.57  0.66 -0.31 -0.31
## indus    0.41 -0.53  1.00  0.06  0.76 -0.39  0.64 -0.71  0.60  0.72
## chas    -0.06 -0.04  0.06  1.00  0.09  0.09  0.09 -0.10 -0.01 -0.04
## nox      0.42 -0.52  0.76  0.09  1.00 -0.30  0.73 -0.77  0.61  0.67
## rm      -0.22  0.31 -0.39  0.09 -0.30  1.00 -0.24  0.21 -0.21 -0.29
## age      0.35 -0.57  0.64  0.09  0.73 -0.24  1.00 -0.75  0.46  0.51
## dis     -0.38  0.66 -0.71 -0.10 -0.77  0.21 -0.75  1.00 -0.49 -0.53
## rad      0.63 -0.31  0.60 -0.01  0.61 -0.21  0.46 -0.49  1.00  0.91
## tax      0.58 -0.31  0.72 -0.04  0.67 -0.29  0.51 -0.53  0.91  1.00
## ptratio  0.29 -0.39  0.38 -0.12  0.19 -0.36  0.26 -0.23  0.46  0.46
## black   -0.39  0.18 -0.36  0.05 -0.38  0.13 -0.27  0.29 -0.44 -0.44
## lstat    0.46 -0.41  0.60 -0.05  0.59 -0.61  0.60 -0.50  0.49  0.54
## medv    -0.39  0.36 -0.48  0.18 -0.43  0.70 -0.38  0.25 -0.38 -0.47
##         ptratio black lstat  medv
## crim       0.29 -0.39  0.46 -0.39
## zn        -0.39  0.18 -0.41  0.36
## indus      0.38 -0.36  0.60 -0.48
## chas      -0.12  0.05 -0.05  0.18
## nox        0.19 -0.38  0.59 -0.43
## rm        -0.36  0.13 -0.61  0.70
## age        0.26 -0.27  0.60 -0.38
## dis       -0.23  0.29 -0.50  0.25
## rad        0.46 -0.44  0.49 -0.38
## tax        0.46 -0.44  0.54 -0.47
## ptratio    1.00 -0.18  0.37 -0.51
## black     -0.18  1.00 -0.37  0.33
## lstat      0.37 -0.37  1.00 -0.74
## medv      -0.51  0.33 -0.74  1.00

Summary of cor_matrix variables.

# center and standardize variables
boston_scaled <- scale(Boston)
# summaries of the scaled variables
summary(boston_scaled)
##       crim                 zn               indus        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202  
##       chas              nox                rm               age         
##  Min.   :-0.2723   Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331  
##  1st Qu.:-0.2723   1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366  
##  Median :-0.2723   Median :-0.1441   Median :-0.1084   Median : 0.3171  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.:-0.2723   3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059  
##  Max.   : 3.6648   Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164  
##       dis               rad               tax             ptratio       
##  Min.   :-1.2658   Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047  
##  1st Qu.:-0.8049   1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876  
##  Median :-0.2790   Median :-0.5225   Median :-0.4642   Median : 0.2746  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6617   3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058  
##  Max.   : 3.9566   Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372  
##      black             lstat              medv        
##  Min.   :-3.9033   Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.: 0.2049   1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median : 0.3808   Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.4332   3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 0.4406   Max.   : 3.5453   Max.   : 2.9865

After scaling the variables, the means of each variables have become very close to zero. Many of the minimum and maximum values for each variable have decreased.

Next we place the variables into quantiles, to get high, med_low, med_high, and low rates of crime into their own categories

# class of the boston_scaled object
class(boston_scaled)
## [1] "matrix"
# change the object to data frame
boston_scaled<-as.data.frame(boston_scaled)

# create a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)

bins
##           0%          25%          50%          75%         100% 
## -0.419366929 -0.410563278 -0.390280295  0.007389247  9.924109610

First, we use the quantiles; low, med_low, med_high, and high as the break points and create a categorical variable of the crime rate in the Boston dataset.

# create a categorical variable 'crime'
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, labels = c("low", "med_low", "med_high", "high"))

# look at the table of the new factor crime
table(crime)
## crime
##      low  med_low med_high     high 
##      127      126      126      127
# summary of the scaled crime rate
summary(crime)
##      low  med_low med_high     high 
##      127      126      126      127

Then the old crime rate variable is removed and then the dataset is divided into train and test so that 80% of the data belongs to the train set.

# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)

# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)
# boston_scaled is available

# number of rows in the Boston dataset 
n <- nrow(boston_scaled)

# choose randomly 80% of the rows
ind <- sample(n,  size = n * 0.8)

# create train set
train <- boston_scaled[ind,]

# create test set 
test <- boston_scaled[-ind,]

# save the correct classes from test data
correct_classes <- test$crime

# remove the crime variable from test data
test <- dplyr::select(test, -crime)

Linear discriminant analysis

Linear Discriminant analysis is a classification method that finds the linear combination of the variables that separate the target variable classes.

Here I fit the linear discriminant analysis on the train set. Use the categorical crime rate as the target variable and all the other variables in the dataset as predictor variables. Drew the LDA (bi)plot.

# linear discriminant analysis
lda.fit <- lda(crime~., data = train)

# print the lda.fit object
lda.fit
## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2574257 0.2475248 0.2574257 0.2376238 
## 
## Group means:
##                  zn      indus        chas        nox         rm
## low       0.8233984 -0.8895665 -0.08304540 -0.8379681  0.4541814
## med_low  -0.1129228 -0.2630300 -0.11484506 -0.5464043 -0.1847157
## med_high -0.3948894  0.2036437  0.25766519  0.3599381  0.1318973
## high     -0.4872402  1.0149946 -0.06727176  1.0453080 -0.3962699
##                 age        dis        rad        tax     ptratio
## low      -0.8430078  0.7694689 -0.6969630 -0.7728076 -0.42049155
## med_low  -0.3130465  0.3387163 -0.5443053 -0.4326475 -0.01826078
## med_high  0.4307150 -0.3697316 -0.4076377 -0.2979065 -0.22329348
## high      0.8042841 -0.8581542  1.6596029  1.5294129  0.80577843
##               black       lstat        medv
## low       0.3764367 -0.77932092  0.53914143
## med_low   0.3192487 -0.07258910 -0.05782324
## med_high  0.1047326 -0.01374577  0.18137831
## high     -0.8561780  0.88516242 -0.70396865
## 
## Coefficients of linear discriminants:
##                 LD1         LD2         LD3
## zn       0.12758506  0.60745738 -0.94286048
## indus    0.04754003 -0.24316240  0.23844737
## chas    -0.06671071 -0.11570054 -0.04217161
## nox      0.22797702 -0.91556192 -1.27400735
## rm      -0.13707307 -0.08585354 -0.19923329
## age      0.27887182 -0.33809304 -0.13218378
## dis     -0.07834757 -0.35002075  0.19922148
## rad      3.43604981  0.94544336 -0.36520923
## tax      0.01850730 -0.01161153  0.83765108
## ptratio  0.11458152 -0.01320203 -0.26424769
## black   -0.16201856  0.01586183  0.14993926
## lstat    0.16433729 -0.23040505  0.51472109
## medv     0.16811060 -0.40727299 -0.05838591
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9537 0.0332 0.0131
# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

Next I saved the crime categories from the test set and then remove the categorical crime variable from the test dataset. Then I predicted the classes with the LDA model on the test data. I Cross tabulated the results with the crime categories from the test set. Commented on the results.

# target classes as numeric
classes <- as.numeric(train$crime)

# plot the lda results
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 1)

# lda.fit, correct_classes and test are available

# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)

# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)
##           predicted
## correct    low med_low med_high high
##   low       20       3        0    0
##   med_low    6      16        4    0
##   med_high   1       6       14    1
##   high       0       0        1   30
# lda.fit, correct_classes and test are available

# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)

# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class) %>% addmargins
##           predicted
## correct    low med_low med_high high Sum
##   low       20       3        0    0  23
##   med_low    6      16        4    0  26
##   med_high   1       6       14    1  22
##   high       0       0        1   30  31
##   Sum       27      25       19   31 102
# Percentage inaccuracy
table(correct = correct_classes, predicted = lda.pred$class) %>% prop.table() %>% addmargins()
##           predicted
## correct            low     med_low    med_high        high         Sum
##   low      0.196078431 0.029411765 0.000000000 0.000000000 0.225490196
##   med_low  0.058823529 0.156862745 0.039215686 0.000000000 0.254901961
##   med_high 0.009803922 0.058823529 0.137254902 0.009803922 0.215686275
##   high     0.000000000 0.000000000 0.009803922 0.294117647 0.303921569
##   Sum      0.264705882 0.245098039 0.186274510 0.303921569 1.000000000
loss_func <- function(class, prob) {
  n_wrong <- class != prob
  mean(n_wrong)
}
loss_func(class = correct_classes, prob = lda.pred$class)
## [1] 0.2156863

From the cross table, it is shown that most of the predictions are correct and from the total proportion of inaccurately predicted individuals the prediction model is close to 31% incorrect, making the model close to 69% accurate. This is a relatively good model based on these results.

Here I reloaded the Boston dataset and standardize the dataset (we did not do this in the Datacamp exercises, but you should scale the variables to get comparable distances). Calculated the distances between the observations.

# load MASS and Boston
library(MASS)
data('Boston')

#scaled
boston_scaled=scale(Boston)

# euclidean distance matrix
dist_eu <- dist(boston_scaled)

# look at the summary of the distances
summary(dist_eu)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1343  3.4625  4.8241  4.9111  6.1863 14.3970
# manhattan distance matrix
dist_man <- dist(boston_scaled, method = 'manhattan')

# look at the summary of the distances
summary(dist_man)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.2662  8.4832 12.6090 13.5488 17.7568 48.8618
summary(dist_eu)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1343  3.4625  4.8241  4.9111  6.1863 14.3970

Ran the k-means algorithm on the dataset. Investigated what is the optimal number of clusters and reran the algorithm again. Visualized the clusters (for example with the pairs() or ggpairs() functions, where the clusters are separated with colors) and interpret the results.

# k-means clustering
km <-kmeans(boston_scaled, centers = 3)

# plot the Boston dataset with clusters
pairs(boston_scaled, col = km$cluster)

# MASS, ggplot2 and Boston dataset are available
set.seed(123)

# determine the number of clusters
k_max <- 10

# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})

# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')

# k-means clustering
km <-kmeans(boston_scaled, centers = 2)

# plot the Boston dataset with clusters
pairs(boston_scaled, col = km$cluster)

The graph shows that 2 clusters are the optimal number of clusters. The total of within cluster sum of squares (WCSS) vs. the number of cluster changes is plotted. We can observe the clusters on the data and how they relate in combination for each variable.

Bonus:

Perform k-means on the original Boston data with some reasonable number of clusters (> 2). Remember to standardize the dataset. Then perform LDA using the clusters as target classes. Include all the variables in the Boston data in the LDA model. Visualize the results with a biplot (include arrows representing the relationships of the original variables to the LDA solution).

library(MASS)
data('Boston')

#scaled
boston_scaledx=scale(Boston)
km2 <-kmeans(boston_scaledx, centers = 3)
boston_scaledx <- as.data.frame(boston_scaledx)
#LDA
lda.fit2 <- lda(km2$cluster ~., data=boston_scaledx)

#lda biplot 
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

# target classes as numeric
classes <- as.numeric(km2$cluster)

# plot the lda results
plot(lda.fit2, dimen = 2, col=classes, pch= classes)
lda.arrows(lda.fit, myscale = 1)

It appears that rad (index of accessibility to radial highways) is the most influencial linear separators for the clusters, as well as nox (nitrogen oxides concentration) and (zn) proportion of residential land zoned for lots over 25,000 sq.ft. #Super Bonus

#First run the code below for the (scaled) train data that you used to fit the LDA. The code creates a matrix product, which is a projection of the data points.
model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)
## [1] 404  13
dim(lda.fit$scaling)
## [1] 13  3
# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)

library(plotly)
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:MASS':
## 
##     select
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers')
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = train$crime)

The first graphic is not that clear. There are two clusters we can identify which observations belong to which crime classes.

In the second graphic we observe the graphic where there are 4 colors, one color for each cluster; green is low, med_low is orange, blue is med_high, pink is high. The optimal cluster number is 2 clusters.

Chapter 5: Dimensionality reduction techniques

human = read.table('/Users/eric/Documents/GitHub/IODS-project/data/human.txt',sep='\t',header = TRUE)

str(human)
## 'data.frame':    155 obs. of  8 variables:
##  $ Edu2.FM  : num  1.007 0.997 0.983 0.989 0.969 ...
##  $ Labo.FM  : num  0.891 0.819 0.825 0.884 0.829 ...
##  $ Life.Exp : num  81.6 82.4 83 80.2 81.6 80.9 80.9 79.1 82 81.8 ...
##  $ Edu.Exp  : num  17.5 20.2 15.8 18.7 17.9 16.5 18.6 16.5 15.9 19.2 ...
##  $ GNI      : int  64992 42261 56431 44025 45435 43919 39568 52947 42155 32689 ...
##  $ Mat.Mor  : int  4 6 6 5 6 7 9 28 11 8 ...
##  $ Ado.Birth: num  7.8 12.1 1.9 5.1 6.2 3.8 8.2 31 14.5 25.3 ...
##  $ Parli.F  : num  39.6 30.5 28.5 38 36.9 36.9 19.9 19.4 28.2 31.4 ...
dim(human)
## [1] 155   8

The human data is from the UNITED NATIONS DEVELOPMENT PROGRAMME Human Development Reports 2016

“The HDI(human development index) was created to show that people and their abilities should be the ultimate criteria for assessing the development of a country, not economic growth alone. The HDI has the potential to also be used to question national policy choices, asking how two countries with the same level of GNI per capita can end up with different human development outcomes.” The data.frame has 155 obs. of 8 variables:

Accessed libraries

library(MASS)
library(corrplot)
library(tidyr)
library(GGally)
library(ggplot2)

Visualized human

ggpairs(human)

cor_matrix=cor(human)
corrplot(cor_matrix,type='upper')

From the scatter plot, it is seen that Edu.Exp and Life.Exp have a strong postivie correlation (0.789) as well as Mat.Mor and Ado.Birth. While Life.Exp and Mat.Mor have a strong negative correlation(-0.857).

In the corrplot Positive correlations are displayed in blue and negative correlations in red color. Color intensity and the size of the circle are proportional to the correlation coefficients. In the right side of the correlogram, the legend color shows the correlation coefficients and the corresponding colors. The corrplot shows the correlations between variables of the Boston dataset. The highest postive correlation can be seen between Life.Exp and Edu.Exp and the highest negative correlatiuon is seen between Life.Exp and Math.Mor

Perform Principal Component Analysis (PCA) non-standardized variables

pca_human <- prcomp(human)
s <- summary(pca_human)
s
## Importance of components:
##                              PC1      PC2   PC3   PC4   PC5   PC6    PC7
## Standard deviation     1.854e+04 185.5219 25.19 11.45 3.766 1.566 0.1912
## Proportion of Variance 9.999e-01   0.0001  0.00  0.00 0.000 0.000 0.0000
## Cumulative Proportion  9.999e-01   1.0000  1.00  1.00 1.000 1.000 1.0000
##                           PC8
## Standard deviation     0.1591
## Proportion of Variance 0.0000
## Cumulative Proportion  1.0000
pca_pr=round(100*s$importance[2,],digits=1)
pc_lab=paste0(names(pca_pr),'(',pca_pr,'%)')
biplot(pca_human,choices=1:2,cex=c(0.8,1),xlab=pc_lab[1],ylab=pc_lab[2])
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped

With the non-standardized variables, it is difficult to read the bi-plot. The scales are extreme (-200000 and -0.4). PC1 is 100% and PC2 is 0%

Perform Principal Component Analysis (PCA) on standardized variables

human_std <- scale(human)
pca_human=prcomp(human_std)
s=summary(pca_human)
s
## Importance of components:
##                           PC1    PC2     PC3     PC4     PC5     PC6
## Standard deviation     2.0708 1.1397 0.87505 0.77886 0.66196 0.53631
## Proportion of Variance 0.5361 0.1624 0.09571 0.07583 0.05477 0.03595
## Cumulative Proportion  0.5361 0.6984 0.79413 0.86996 0.92473 0.96069
##                            PC7     PC8
## Standard deviation     0.45900 0.32224
## Proportion of Variance 0.02634 0.01298
## Cumulative Proportion  0.98702 1.00000
pca_pr=round(100*s$importance[2,],digits=1)
pc_lab=paste0(names(pca_pr),'(',pca_pr,'%)')
biplot(pca_human,choices = 1:2,cex=c(0.8,1),xlab=pc_lab[1],ylab=pc_lab[2])

The biplots between standardized and nonstandardized variables have different results because the biplot with the non-standardized variable had extreme variance while the biplot wtih standardized variables, all of the variables have similiar variance. It is easy to see that between the two biplots, there is a big difference between the distribution of the data.

PC1 is 53.6% and PC2 is 16.2%.

It is clearly show in the biplot that Mat.Mor and Ado.Birth have a positive relationship between each other. It is also show that Edu.Exp,Edu2.FM,GNI and Life.Exp have a positive relationship between other.

While Palri.f and Labo.fm have have a negative relationship.

Loaded Libraries

library(FactoMineR)
library(ggplot2) 
library(dplyr) 
## 
## Attaching package: 'dplyr'
## The following object is masked from 'package:MASS':
## 
##     select
## The following object is masked from 'package:GGally':
## 
##     nasa
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library(tidyr)

Explored data set tea

data('tea')
str(tea)
## 'data.frame':    300 obs. of  36 variables:
##  $ breakfast       : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
##  $ tea.time        : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
##  $ evening         : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
##  $ lunch           : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
##  $ dinner          : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
##  $ always          : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
##  $ home            : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
##  $ work            : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
##  $ tearoom         : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
##  $ friends         : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
##  $ resto           : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
##  $ pub             : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
##  $ Tea             : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
##  $ How             : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
##  $ sugar           : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
##  $ how             : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ where           : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ price           : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
##  $ age             : int  39 45 47 23 48 21 37 36 40 37 ...
##  $ sex             : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
##  $ SPC             : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
##  $ Sport           : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
##  $ age_Q           : Factor w/ 5 levels "15-24","25-34",..: 3 4 4 1 4 1 3 3 3 3 ...
##  $ frequency       : Factor w/ 4 levels "1/day","1 to 2/week",..: 1 1 3 1 3 1 4 2 3 3 ...
##  $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
##  $ spirituality    : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
##  $ healthy         : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
##  $ diuretic        : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
##  $ friendliness    : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
##  $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
##  $ feminine        : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
##  $ sophisticated   : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
##  $ slimming        : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ exciting        : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
##  $ relaxing        : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
##  $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...
dim(tea)
## [1] 300  36

The data set tea contains 300 observations and 36 variables, many of them are factors.

Created and visualized tea_time

keep_columns <- c("Tea", "How", "how", "sugar", "where", "lunch")
tea_time <- dplyr::select(tea, one_of(keep_columns))
summary(tea_time)
##         Tea         How                      how           sugar    
##  black    : 74   alone:195   tea bag           :170   No.sugar:155  
##  Earl Grey:193   lemon: 33   tea bag+unpackaged: 94   sugar   :145  
##  green    : 33   milk : 63   unpackaged        : 36                 
##                  other:  9                                          
##                   where           lunch    
##  chain store         :192   lunch    : 44  
##  chain store+tea shop: 78   Not.lunch:256  
##  tea shop            : 30                  
## 
str(tea_time)
## 'data.frame':    300 obs. of  6 variables:
##  $ Tea  : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
##  $ How  : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
##  $ how  : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ sugar: Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
##  $ where: Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ lunch: Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
gather(tea_time) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar() + theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))
## Warning: attributes are not identical across measure variables;
## they will be dropped

Created tea_time from tea by selecting the columns “Tea”, “How”, “how”, “sugar”, “where”, “lunch”. In Lunch, Not Lunch has a higher count then lunch. In sugar, No sugar’s count is close to sugar. In where, chain store has the highest count. In How, alone has the highest count. In tea, earl grey has the highest count. In how, tea bag has the highest count.

Performed MCA on tea_time

mca=MCA(tea_time,graph=FALSE)
summary(mca)
## 
## Call:
## MCA(X = tea_time, graph = FALSE) 
## 
## 
## Eigenvalues
##                        Dim.1   Dim.2   Dim.3   Dim.4   Dim.5   Dim.6
## Variance               0.279   0.261   0.219   0.189   0.177   0.156
## % of var.             15.238  14.232  11.964  10.333   9.667   8.519
## Cumulative % of var.  15.238  29.471  41.435  51.768  61.434  69.953
##                        Dim.7   Dim.8   Dim.9  Dim.10  Dim.11
## Variance               0.144   0.141   0.117   0.087   0.062
## % of var.              7.841   7.705   6.392   4.724   3.385
## Cumulative % of var.  77.794  85.500  91.891  96.615 100.000
## 
## Individuals (the 10 first)
##                       Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3
## 1                  | -0.298  0.106  0.086 | -0.328  0.137  0.105 | -0.327
## 2                  | -0.237  0.067  0.036 | -0.136  0.024  0.012 | -0.695
## 3                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 4                  | -0.530  0.335  0.460 | -0.318  0.129  0.166 |  0.211
## 5                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 6                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 7                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 8                  | -0.237  0.067  0.036 | -0.136  0.024  0.012 | -0.695
## 9                  |  0.143  0.024  0.012 |  0.871  0.969  0.435 | -0.067
## 10                 |  0.476  0.271  0.140 |  0.687  0.604  0.291 | -0.650
##                       ctr   cos2  
## 1                   0.163  0.104 |
## 2                   0.735  0.314 |
## 3                   0.062  0.069 |
## 4                   0.068  0.073 |
## 5                   0.062  0.069 |
## 6                   0.062  0.069 |
## 7                   0.062  0.069 |
## 8                   0.735  0.314 |
## 9                   0.007  0.003 |
## 10                  0.643  0.261 |
## 
## Categories (the 10 first)
##                        Dim.1     ctr    cos2  v.test     Dim.2     ctr
## black              |   0.473   3.288   0.073   4.677 |   0.094   0.139
## Earl Grey          |  -0.264   2.680   0.126  -6.137 |   0.123   0.626
## green              |   0.486   1.547   0.029   2.952 |  -0.933   6.111
## alone              |  -0.018   0.012   0.001  -0.418 |  -0.262   2.841
## lemon              |   0.669   2.938   0.055   4.068 |   0.531   1.979
## milk               |  -0.337   1.420   0.030  -3.002 |   0.272   0.990
## other              |   0.288   0.148   0.003   0.876 |   1.820   6.347
## tea bag            |  -0.608  12.499   0.483 -12.023 |  -0.351   4.459
## tea bag+unpackaged |   0.350   2.289   0.056   4.088 |   1.024  20.968
## unpackaged         |   1.958  27.432   0.523  12.499 |  -1.015   7.898
##                       cos2  v.test     Dim.3     ctr    cos2  v.test  
## black                0.003   0.929 |  -1.081  21.888   0.382 -10.692 |
## Earl Grey            0.027   2.867 |   0.433   9.160   0.338  10.053 |
## green                0.107  -5.669 |  -0.108   0.098   0.001  -0.659 |
## alone                0.127  -6.164 |  -0.113   0.627   0.024  -2.655 |
## lemon                0.035   3.226 |   1.329  14.771   0.218   8.081 |
## milk                 0.020   2.422 |   0.013   0.003   0.000   0.116 |
## other                0.102   5.534 |  -2.524  14.526   0.197  -7.676 |
## tea bag              0.161  -6.941 |  -0.065   0.183   0.006  -1.287 |
## tea bag+unpackaged   0.478  11.956 |   0.019   0.009   0.000   0.226 |
## unpackaged           0.141  -6.482 |   0.257   0.602   0.009   1.640 |
## 
## Categorical variables (eta2)
##                      Dim.1 Dim.2 Dim.3  
## Tea                | 0.126 0.108 0.410 |
## How                | 0.076 0.190 0.394 |
## how                | 0.708 0.522 0.010 |
## sugar              | 0.065 0.001 0.336 |
## where              | 0.702 0.681 0.055 |
## lunch              | 0.000 0.064 0.111 |
plot(mca,invisible=c('ind'),habillage='quali')

From the summary; tea bag, earl grey, and lemon are most likely Dim3. lone, green, milk, black tea, tea bag+unpackaged, unpackaged, and other are most likely Dim1. It seems that lunch, How, Tea, and sugar are most likely in Dim3. where and how are most likely Dim.1 while where could also be Dim2.

MCA is generally used to analyse a data set from survey. The goal is to identify a group of individuals with similar profile in their answers to the questions and the associations between variable categories From the MCA graph, it is clear that tea bag + unpackaged is the most associated with Dim1 while other is the most associated with Dim2.